Variational Geometry

In the study of ‘variations’, constraints can present a major complication. Before the effects of variations can be ascertained it may be necessary to determine the directions in which something can be varied at all. This may be difficult, whether the variations are aimed at tests of optimality or stability, or arise in trying to understand the consequences of perturbations in the data parameters on which a mathematical model might depend.

In maximizing or minimizing a function over a set C ⊂ ℝⁿ, for instance, properties of the boundary of C can be crucial in characterizing a solution. When C is specified by a system of constraints such as inequalities, however, the boundary may have all kinds of curvilinear facets, edges and corners. Standard methods of geometric analysis can't cope with such a lack of smoothness except in simple cases where the pieces making up the boundary of C are neatly laid out and can be dealt with one by one.